Find the exact solution of the equation in its simplest form: 3^x * e^4x = e^7.

First take natural logs of both sides, giving us: ln(3^x * e^4x) = ln(e⁷).  The RHS can be simplified using the definition of natural logs, so ln(e⁷) = 7.

We can then apply the product rule for logs (ln(a * b) = ln(a) + ln(b)), which gives: ln(3^x) + ln(e^4x) = 7.

Using the power rule, (ln(a^b) = b*ln(a)) and the def. of natural logs, the equation can be simplified further: xln(3) + 4x = 7.

Factorise by taking out the factors of x to give: x(ln(3) + 4) = 7.

Then divide both sides by (ln(3) + 4) to get the equation with x as the subject: x = 7 / (ln(3) + 4).

We now have an exact value for x and we can check the answer by substituting it back into the original equation and checking that we get e⁷.